Are you using IMAP or POP email? Sounds like you have a POP account or have the account entered in Thunderbird as POP since POP accounts only connect to the Inbox of the email server. You have to use IMAP email to see and sync all folders between multiple devices.

Thanks, very helpful indeed - sounds like it may be the solution. But how do I find out whether it's POP or IMAP? In Thunderbird, under 'Server Settings' the Hotmail account is defined as POP (pop-mail.outlook.com). Can I just manually replace that with something like IMAP-mail.outlook.com??

You need to add your account to Thunderbird all over again. There is no way in any mail client to convert an account from POP to IMAP

Many thanks, really appreciate the help. However, I now worry that the synchronisation with IMAP means that I cannot delete e-mails on the server while keeping them on my PC. I am uncomfortable leaving all of my e-mails in the 'cloud' and occasionally wish to delete them from the Hotmail server. CAn that be done while keeping them on my PC?

Thunderbird provides the Local Folders account for this. This special account stores its contents on your local system (usually your hard disk) and so is independent of any online servers. Move messages there and they are yours to keep.